Question

In Problem and , find the indicated Fourier series. Then differentiate your result repeatedly (both the function and the series) until you get a discontinuous function. Use a computer to plot f(x)and the derivative functions. For each graph, plot on the same axes one or more terms of the corresponding Fourier series. Note the number of terms needed for a good fit (see comment at the end of the section).

Short answer

The Fourier series of the function and up to the third derivative (when the function becomes discontinuous) are

Step-by-step solution

Given information

The given function is

Definition of Fourier series

A Fourier series is that a sum that be a periodic function as a sum of sine and cosine waves. It can be written as

The corresponding Fourier coefficients are

Find the derivatives of the given function

The function f is differentiable at a, the limits L exists, then this limits is called the derivatives of f at a and denoted as f^'(a)

The function and its derivatives are

Plot the graph of the function and its derivatives

The graph of the function and its derivatives

Evaluate the value of coefficients

The function is even, so b_n=0 and the other coefficients are:

Further, Solve for the value of a_n

Thus, the Fourier series of the function and up to the third derivative (when the function becomes discontinuous) are